The rigidity and global rigidity of bar-joint frameworks in Euclidean spaces has been intensely studied in recent years (e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}) and has a rich history going as far back as classical work of Euler and Cauchy on Euclidean polyhedra.
In the last decade, work on rigidity has been generalised to various non-Euclidean normed spaces (e.g. [7]}, [8]}, [9]}, [10]}, [11]}). All of these results concern characterising the combinatorial nature of the `generic' behaviour of frameworks. This article extends this to frameworks with two points lying in the same location. The difficulty that already arises in this context shows how necessary the genericity assumption in those papers really was.
Frameworks with coincident points have been considered in the Euclidean context [12]}, [13]} and applied to global rigidity there [14]}, as well as for frameworks on surfaces [15]}.
